The document discusses transfer functions and Laplace transforms. It explains that transfer functions provide a simple way to relate the input and output of a system by describing the relationship as a ratio of outputs to inputs in the Laplace domain. Common examples of transfer functions for different system elements are provided, such as gears, amplifiers, and DC motors. It also discusses how to calculate overall transfer functions for systems consisting of multiple elements in series and systems with feedback loops.

1. Week 10 Data Presentation System
Part 2
Mechatronics System Design
Prof. CHARLTON S. INAO
Defence Engineering College,
Debre Zeit , Ethiopia
Transfer Function

2. Transfer Function

3. Comments on Transfer Function

4. Comments on Transfer Function

5. Transfer function
Pierre-Simon Laplace (1749–
1827).
The relationship between the output and the input for
elements used in control systems is frequently
described by a differential equation. However, in
order to make life simple, what we really need is a
simpler relationship than a differential equation giving
the relationship between input and output for a
system, even when the output varies with time. It is
nice and simple to say that the output is just ten times
the input and so describe the system by gain = 10.
But it is not so simple when the relationship between
the input and output is described by a differential
equation.
However, there is a way we can have such a simple form of relationship
where the relationship involves time but it involves writing inputs and
outputs in a different form. It is called the Laplace transform. In this
chapter we will consider how we can carry out such transformations, but
not the mathematics to justify why we can do it; the aim is to enable you
to use the transform as a tool to carry out tasks.

6. In general, when we consider inputs and
outputs of systems as functions of time
then the relationship between the output
and input is given by a differential
equation.
If we have a system composed of two
elements in series with each having its
input-output relationships described by a
differential equation, it is not easy to see
how the output of the system as a whole is
related to its input.

7. • There is a way we can overcome this problem and
that is to transform the differential equations into a
more convenient form by using the Laplace.
• This form is a much more convenient way of
describing the relationship than a differential
equation since it can be easily manipulated by the
basic rules of algebra.

8. To carry out the transformation we follow the
following rules:

17. Transfer Function
• The term gain to relate the input and output of a
system with gain G = output/input When we are
working with inputs and outputs described as
functions of s we define the transfer function G(s)
as [output Y(s)/input X(s)] when all initial
conditions before we apply the input are zero:

18. Transfer function as
the factor that
multiplies the input to
give the output.
A transfer function can be represented as a block
diagram with X(s) the input, Y(s) the output and the
transfer function G(s) as the operator in the box that
converts the input to the output. The block
represents a multiplication for the input. Thus, by
using the Laplace transform of inputs and outputs,
we can use the transfer function as a simple
multiplication factor, like the gain discussed
previously.

21. Transfer functions of common
system elements
• By considering the relationships between
the inputs to systems and their outputs we
can obtain transfer functions for them and
hence describe a control system as a series
of interconnected blocks, each having its
input-output characteristics defined by a
transfer function. The following are transfer
functions which are typical of commonly
encountered system elements:

22. 1 Gear train
For the relationship between the input speed and output speed with a
gear train having a gear ratio N:
transfer function = N
2 Amplifier
For the relationship between the output voltage and the input
voltage with G as the constant gain:
transfer function = G
3 Potentiometer
For the potentiometer acting as a simple potential divider circuit
the relationship between the output voltage and the input
voltage is the ratio of the resistance across which the output is
tapped to the total resistance across which the supply voltage is
applied and so is a constant and hence the transfer function is a
constant K:
transfer function = K

23. 4 Armature-controlled dc. motor
For the relationship between the drive shaft speed and the input voltage
to the armature is:
where L represents the inductance of the armature circuit and R its
resistance.
This was derived by considering armature circuit as effectively inductance
in series with resistance and hence:
and so, with no initial conditions:
and, since the output torque is proportional to the armature current we have
a transfer function of the form

24. 5 .Valve controlled hydraulic actuator
The output displacement of the hydraulic cylinder is
related to the input displacement of the valve shaft by a
transfer function of the form:
6. Heating system
The relationship between the resulting temperature
and the input to a heating element is typically of the
form:
where C is a constant representing the thermal capacity of
the system and R a constant representing its thermal
resistance.

25. 7.Tachogenerator
The relationship between the output voltage and the input
rotational speed is likely to be a constant K and so
represented by:
transfer function = K
8 Displacement and rotation
For a system where the input is the rotation of a shaft
and the output, as perhaps the result of the rotation of a
screw, a displacement, since speed is the rate of
displacement we have v = dy/dt and so V(s) = sY(s) and
tlie transfer function is:

26. 9 Height of liquid level in a container
The height of liquid in a container depends on the rate at
which liquid enters the container and the rate at which it is
leaving. The relationship between the input of the rate of
liquid entering and the height of liquid in the container is of
the form:
where A is the constant cross-sectional area of the
container, p the density of the liquid, g the acceleration
due to gravity and R the hydraulic resistance offered by
the pipe through which the liquid leaves the container.

27. Illustration of transfer function of
common system elements

31. DC Motor Electrical Diagram and
Sketch

35. Transfer functions and systems
 Consider a speed control system involving a differential amplifier to amplify
the error signal and drive a motor, this then driving a shaft via a gear
system. Feedback of the rotation of the shaft is via a tachogenerator.
1 The differential amplifier might be assumed to give an output
directly proportional to the error signal input and so be
represented by a constant transfer function K, i.e. a gain K
which does not change with time.
2 The error signal is an input to the armature circuit of the motor
and results in the motor giving an output torque which is
proportional to the armature current. The armature circuit can be
assumed to be a circuit having inductance L and resistance R
and so a transfer function of

36. 3 The torque output of the motor is transformed to rotation of the drive
shaft by a gear system and we might assume that the rotational speed
is proportional to the input torque and so represent the transfer
function of the gear system by a constant transfer function N, i.e. the
gear ratio.
4 The feedback is via a tachogenerator and we might make the
assumption that the output of the generator is directly proportional to its
input and so represent it by a constant transfer function H.
The block diagram of the control system might thus be like:
Block diagram for the control system for speed of a shaft with the terms in the
boxes being the transfer functions for the elements concerned

37. System transfer functions
Consider the overall transfer functions of
systems involving series connected
elements and systems with feedback loops.
Systems in series
Consider a system of two subsystems in series
The first subsystem has an input of X(s) and an output of Y1(s); thus, G1(s) =
Y1 (s)/X(s). The second subsystem has an input of Y1 (s) and an output of
Y(s) ;thus, G2(s) = Y(s)/Y1(s)

38. We thus have:

40. Systems with feedback
• For systems with a negative feedback loop we can have the situation
shown in Figure below where the output is fed back via a system with a
transfer function H(s) to subtract from the input to the system G(s). The
feedback system has an input of Y(s) and thus an output of H(s)Y(s). Thus
the feedback signal is H(s)Y(s).
System
with
negative
feedback
The error is the difference between the system input signal X(s)
and the feedback signal and is thus:

41. This error signal is the input to the G(s) system and gives an
output of Y(s). Thus:
and so:
which can be rearranged to give
For a system with a negative feedback, the overall transfer
function is the forward path transfer function divided by one
plus the product of the forward path and feedback path
transfer functions.

42. For a system with positive
feedback (Figure at the right), the
feedback signal is H(s)Y(s) and
thus the input to the G(s) system
is X(s) + H(s)Y(s). Hence:
and so:
This can be
rearranged to
give:
For a system with a positive feedback, the overall transfer function is the
forward path transfer function divided by one minus the product of the
forward path and feedback path transfer functions.

43. Example
Determine the overall transfer function for a control system (Figure)
which has a negative feedback loop with a transfer function 4
and a forward path transfer function of 2/(s + 2).
The overall transfer function of the system is:

44. Example
Determine the overall transfer function for a system (Figure) which has
a positive feedback loop with a transfer function 4 and a forward path
transfer function of 2/(5 + 2).
The overall transfer function is:

45. Block manipulation
Very often, systems may have many elements and
sometimes more than one input. A single input-single
output system is often termed a SISO
system while a multiple input-multiple output
system is a MISO system.
The following are some of the ways we can
reorganize the blocks in a block diagram of a
system in order to produce simplification and still
give the same overall transfer function for the
system. To simplify the diagrams, the (s) has been
omitted; it should, however, be assumed for all
dynamic situations.

46. Blocks in series
As indicated in Section: System series , Figure
below shows the basic rule for simplifying blocks in
series.

47. Moving takeoff points
As a means of simplifying block diagrams it is often
necessary to move takeoff points. The following figures
give the basic rules for such movements.
Moving a takeoff point to beyond a block
Moving a takeoff point to ahead of a block

48. Moving a summing point
As a means of simplifying block diagrams it is often necessary
to move summing points. The following figures give the basic
rules for such movements.
Rearrangement of summing points
Interchange of summing points

49. Moving a summing point ahead of a block
Moving a summing point beyond a block

50. Changing feedback and forward paths
Figures below show block simplification techniques when changing
feed forward and feedback paths.
Removing a block from a feedback path
Removing a block from a forward path

51. Example
Use block simplification techniques to simplify the system shown below

52. 1. Moving
a takeoff
point
2. Eliminating
a feed
forward loop
3. Simplifying
series
elements

53. 4. Simplifying
a feedback
element
5. Simplifying
series
elements
6. Simplifying
negative
feedback

54. ADDITIONAL
APPLICATIONS

59. Example 1

61. Example 2
Example 3

62. Example 4
Example 5

63. Example 5. Convert the differential equation to a transfer function
Answer:
Exercises Class participation

66. ADDITIONAL
EXERCISES

70. Find the transfer function of
the electrical network shown
in phase lead form.

72. Find the transfer
function of the
electrical network
shown
Assuming no external load
Applying Kirchoff’s law to electrical network
Taking Laplace transform
putting

73. Redrawing the
figure after
substituting the
values
Find the transfer
function of the
electrical network
shown
Solution
Let

74. Substituting the value of I1(s)
in equation (1)
But from equation (3)
Therefore
Or
Substituting the value of Z1, Z2, Z3 and Z4

77. Transfer Function
where
Also
or
or
when

78. Assume current distribution as shown

81. Write the differential equations for the
electrical shown
Assuming current
distribution as
shown in figure,
the differential
equation are
obtained by the
use of Kirchoff’s
law

82. Determine the transfer
function relation Vo(s) to
Vi(s) for the network
shown
Transfer function is

84. Redrawing the circuit
diagram as shown
and applying
Kirchoff’s law

85. Transfer function is
and
But
Therefore
or

86. Determine the Transfer function of the electrical network
Solution: Assuming current distribution shown, the differential
equations can be written as

95. Answer Key